Wayland, hybrid intel/nvidia, external monitor

Looks normal.
It shows that you are running on the intel by default, and using the nvidia when you use prime-run.

Are they actually the same?
You installed ‘prime’ things and configured your system to work that way?
Or did you just install the nvidia drivers?

As explained above.
You have a hybrid system, with the iGPU used by default. But you are plugging the monitor into the dGPU. Or at least thats what I guess, as thats usually the case in these scenarios.

There are multiple approaches to this like here:

And envycontrol, etc. But you stated you did not want to set it to ‘nvidia-only’

So you either want reverse-prime or something similar

https://wiki.archlinux.org/title/PRIME#Reverse_PRIME

https://wiki.archlinux.org/title/PRIME#Discrete_card_as_primary_GPU